Question: Sides $\overline{AB}$ and $\overline{EF}$ of regular hexagon $ABCDEF$ are extended to meet at point $P$. What is the degree measure of angle $P$?
Explanation: The sum of the angle measures of a hexagon is $180(6-2) = 720$ degrees, so each angle of a regular hexagon measures $720^\circ/6=120^\circ$.  Therefore, $\angle BAF = 120^\circ$, which means $\angle FAP = 180^\circ - \angle BAF = 60^\circ$.  Similarly, $\angle PFA = 60^\circ$.  Since the angles of $\triangle APF$ sum to $180^\circ$, we have $\angle APF = 180^\circ - 60^\circ - 60^\circ = \boxed{60^\circ}$.

[asy]
unitsize(0.6inch);
pair A,B,C,D,EE,F,P;

A = (1,0);
B = rotate(60)*A;
C=rotate(60)*B;
D = rotate(60)*C;
EE = rotate(60)*D;
F = rotate(60)*EE;

P = A + (A - B);
draw (A--B--C--D--EE--F--A--P--F,linewidth(1));
label("$A$",A,NE);
label("$B$",B,N);
label("$C$",C,N);
label("$D$",D,W);
label("$E$",EE,S);
label("$F$",F,S);
label("$P$",P,S);
[/asy]